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Orr L, Roy PN. Operator Formulation of Feynman Path Centroid Dynamics for Rotations. J Phys Chem A 2024; 128:3419-3433. [PMID: 38651978 DOI: 10.1021/acs.jpca.3c08201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/25/2024]
Abstract
An operator formulation of centroid molecular dynamics (CMD) for rotational degrees of freedom is presented. The quasi-density operator concept was introduced by Jang and Voth [J. Chem. Phys 111, 2357 (1999)] and is used to obtain a phase-space mapping without the need for discretized path integrals. The approach allows the calculation of approximate Kubo-transformed time correlation functions. The particle on a ring is chosen as an illustrative example. Numerical results demonstrate that the proposed approach leads to accurate results when compared with exact diagonalization calculations for linear operators. At very low temperatures, it is found that rotational CMD yields results that are in exact agreement with the quantum dynamics of a spin-1 system.
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Affiliation(s)
- Lindsay Orr
- Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - Pierre-Nicholas Roy
- Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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Ture TM, Jang SJ. Simple and General Unitarity Conserving Numerical Real-Time Propagators of the Time-Dependent Schrödinger Equation Based on Magnus Expansion. J Phys Chem A 2024; 128:2871-2882. [PMID: 38564477 DOI: 10.1021/acs.jpca.3c07866] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/04/2024]
Abstract
Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time ab initio quantum dynamics, and open system quantum dynamics.
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Affiliation(s)
- Taner M Ture
- Department of Chemistry and Biochemistry, Queens College, City University of New York, 65-30 Kissena Boulevard, Queens, New York 11367, United States
- PhD Program in Chemistry, Graduate Center of the City University of New York, New York, New York 10016, United States
| | - Seogjoo J Jang
- Department of Chemistry and Biochemistry, Queens College, City University of New York, 65-30 Kissena Boulevard, Queens, New York 11367, United States
- PhD Programs in Chemistry and Physics, Graduate Center of the City University of New York, New York, New York 10016, United States
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Orr L, Hernández de la Peña L, Roy PN. Formulation of state projected centroid molecular dynamics: Microcanonical ensemble and connection to the Wigner distribution. J Chem Phys 2017; 146:214116. [PMID: 28595402 PMCID: PMC5462618 DOI: 10.1063/1.4984229] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2017] [Accepted: 05/15/2017] [Indexed: 11/14/2022] Open
Abstract
A derivation of quantum statistical mechanics based on the concept of a Feynman path centroid is presented for the case of generalized density operators using the projected density operator formalism of Blinov and Roy [J. Chem. Phys. 115, 7822-7831 (2001)]. The resulting centroid densities, centroid symbols, and centroid correlation functions are formulated and analyzed in the context of the canonical equilibrium picture of Jang and Voth [J. Chem. Phys. 111, 2357-2370 (1999)]. The case where the density operator projects onto a particular energy eigenstate of the system is discussed, and it is shown that one can extract microcanonical dynamical information from double Kubo transformed correlation functions. It is also shown that the proposed projection operator approach can be used to formally connect the centroid and Wigner phase-space distributions in the zero reciprocal temperature β limit. A Centroid Molecular Dynamics (CMD) approximation to the state-projected exact quantum dynamics is proposed and proven to be exact in the harmonic limit. The state projected CMD method is also tested numerically for a quartic oscillator and a double-well potential and found to be more accurate than canonical CMD. In the case of a ground state projection, this method can resolve tunnelling splittings of the double well problem in the higher barrier regime where canonical CMD fails. Finally, the state-projected CMD framework is cast in a path integral form.
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Affiliation(s)
- Lindsay Orr
- Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | | | - Pierre-Nicholas Roy
- Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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Welsch R, Song K, Shi Q, Althorpe SC, Miller TF. Non-equilibrium dynamics from RPMD and CMD. J Chem Phys 2016; 145:204118. [DOI: 10.1063/1.4967958] [Citation(s) in RCA: 42] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023] Open
Affiliation(s)
- Ralph Welsch
- Division of Chemistry and Chemical Engineering, California Institute of Technology,1200 E. California Blvd., Pasadena, California 91125, USA
| | - Kai Song
- Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences, Zhongguancun, Beijing 100190, China
| | - Qiang Shi
- Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences, Zhongguancun, Beijing 100190, China
| | - Stuart C. Althorpe
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Thomas F. Miller
- Division of Chemistry and Chemical Engineering, California Institute of Technology,1200 E. California Blvd., Pasadena, California 91125, USA
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Jang S, Sinitskiy AV, Voth GA. Can the ring polymer molecular dynamics method be interpreted as real time quantum dynamics? J Chem Phys 2014. [DOI: 10.1063/1.4870717] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Paesani F, Xantheas SS, Voth GA. Infrared Spectroscopy and Hydrogen-Bond Dynamics of Liquid Water from Centroid Molecular Dynamics with an Ab Initio-Based Force Field. J Phys Chem B 2009; 113:13118-30. [DOI: 10.1021/jp907648y] [Citation(s) in RCA: 114] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Francesco Paesani
- Center for Biophysical Modeling and Simulation and Department of Chemistry, University of Utah, 315 South 1400 East Room 2020, Salt Lake City, Utah 84112-0850, and Chemical and Materials Sciences Division, Pacific Northwest National Laboratory, 902 Battelle Boulevard, P.O. Box 999, MS K1-83, Richland, Washington 99352
| | - Sotiris S. Xantheas
- Center for Biophysical Modeling and Simulation and Department of Chemistry, University of Utah, 315 South 1400 East Room 2020, Salt Lake City, Utah 84112-0850, and Chemical and Materials Sciences Division, Pacific Northwest National Laboratory, 902 Battelle Boulevard, P.O. Box 999, MS K1-83, Richland, Washington 99352
| | - Gregory A. Voth
- Center for Biophysical Modeling and Simulation and Department of Chemistry, University of Utah, 315 South 1400 East Room 2020, Salt Lake City, Utah 84112-0850, and Chemical and Materials Sciences Division, Pacific Northwest National Laboratory, 902 Battelle Boulevard, P.O. Box 999, MS K1-83, Richland, Washington 99352
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Paesani F, Voth GA. The properties of water: insights from quantum simulations. J Phys Chem B 2009; 113:5702-19. [PMID: 19385690 DOI: 10.1021/jp810590c] [Citation(s) in RCA: 189] [Impact Index Per Article: 12.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
The properties of water play a central role in many phenomena of relevance to different areas of science, including physics, chemistry, biology, geology, and climate research. Although well studied for decades, the behavior of water under different conditions and in different environments still remains mysterious and often surprising. In this article, various efforts aimed at providing a comprehensive representation of the water properties at a molecular level through computer modeling and simulation will be described. In particular, the unique role played by the hydrogen-bond network will be examined, first in liquid water, then in the solvation of model biological compounds, and finally in ice, especially highlighting the important effects related to the quantization of the nuclear motion.
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Affiliation(s)
- Francesco Paesani
- Center for Biophysical Modeling and Simulation and Department of Chemistry, University of Utah, 315 South 1400 East Room 2020, Salt Lake City, Utah 84112-0850, USA
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Yoshimori A. New conditions for validity of the centroid molecular dynamics and ring polymer molecular dynamics. J Chem Phys 2008; 128:234105. [DOI: 10.1063/1.2939480] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Jang S. Generalization of the Förster resonance energy transfer theory for quantum mechanical modulation of the donor-acceptor coupling. J Chem Phys 2007; 127:174710. [DOI: 10.1063/1.2779031] [Citation(s) in RCA: 50] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023] Open
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Huh Y, Roy PN. Inclusion of inversion symmetry in centroid molecular dynamics: a possible avenue to recover quantum coherence. J Chem Phys 2006; 125:164103. [PMID: 17092059 DOI: 10.1063/1.2358989] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022] Open
Abstract
Inversion symmetry is included in the operator formulation of the centroid molecular dynamics (CMD). This work involves the development of a symmetry-adapted CMD (SA-CMD), here particularly for symmetrization and antisymmetrization projections. A symmetry-adapted quasidensity operator, as defined by Blinov and Roy [J. Chem. Phys. 115, 7822 (2001)], is employed to obtain the centroid representation of quantum mechanical operators. Numerical examples are given for a single particle confined to one-dimensional symmetric quartic and symmetric double-well potentials. Two SA-CMD simulations are performed separately for both projections, and centroid position autocorrelation functions are obtained. For each projection, the quality of the approximation as well as the accuracy are similar to those of regular CMD. It is shown that individual trajectories from two separate SA-CMD simulations can be properly combined to recover trajectories for Boltzmann statistics. Position autocorrelation functions are compared to the exact quantum mechanical ones. This explicit account of inversion symmetry provides a qualitative improvement on the conventional CMD approach and allows the recovery of some quantum coherence.
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Affiliation(s)
- Yoonjung Huh
- Department of Chemistry, University of Alberta, Edmonton, Alberta T6G 2G2, Canada
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